$D$-elliptic modular curves $X_0(N)$

Source abstract

We present a method for determining whether there exists a degree $D$ rational map from $X_0(N)$ to some elliptic curve $E/\mathbb{Q}$. Moreover, we explain how to obtain a quadratic form that represents all possible degrees of such maps using the degree pairing method. This method previously required odd analytic rank and some additional technical conditions to be satisfied. In this paper, we extended the method to even rank curves as well and remove all technical conditions. We also show how to prove or disprove the existence of degree $D$ maps to elliptic curves by a lifting criterion on the lattices $H_1(-,\mathbb{Z})$ As an application of this method, we determine all $D$-elliptic curves $X_0(N)$ for all $D\leq 100$. Interestingly, in some cases we found degree $D$ rational maps that we could not explain by degeneracy maps, quotient maps, and modular parameterisation.